DMA: Background Information
The revolution in the field of sub-micron particle sizing came with the development of the differential mobility analyzer (DMA) and the theoretical description of its working by Liu, N. Y. H., and Pui, D. Y. H. (1974). A submicron Aerosol Standard and the Primary, Absolute Calibration of the Condensation Nuclei Counter, J. Colloid and interface Sci. 47:155-171 and by Knutson, E. O., and Whitby, K. T. (1975). Aerosol classification by Electrical Mobility: apparatus theory and applications. J. Aerosol Sci. 6:443-451, both hereby incorporated herein by reference. The DMA uses a combination of applied electric and flow fields to classify particles by their electrical mobility and sampling from a port at an appropriate location can result in an output of particles with a narrow range of electrical mobilities. While the initial application of DMAs was primarily for generation of monodisperse particles, the development of the condensation particle counter (CPC) (See Agarwal, J. K. and Sem, G. J. (1980) Continuous flow, single-particle-counting condensation nucleus counter, J. Aerosol Sci. 11:343-357 hereby incorporated herein by reference.) enabled its deployment for particle size distribution measurements. Initial size distribution measurements with the DMA were made with a stepping-mode operation, where the voltages were sequentially stepped to output particles of different mobilities. This technique, called the differential mobility particle sizer (DMPS), required significant “dead-time” between measurements to ensure steady state aerosol sampling. Accurate measurements with the DMPS required ˜20 minutes to cover a broad range of mobilities.
Wang and Flagan (See Wang, S. C., and Flagan, R. C. (1990) Scanning Electrical Mobility Spectrometer, Aerosol Sci. Technol. 13: 230-240 hereby incorporated herein by reference) showed that an exponential voltage ramp can be used to speed up measurements, and for this operation the shape of transfer function remains unchanged throughout the scan. This instrument, known as Scanning Electrical Mobility Analyzer (SEMS), decreased measurement time to ˜5 minutes. Faster measurements with SEMS are complicated by the effect of particle concentration smearing in the CPC and plumbing delays between the DMA and the CPC. There have been several approaches to account for these non-idealities. The effect of smearing and plumbing delay on transfer function was considered in Russell et al. (See Russell, L. M., Flagan, R. C., and Seinfeld, J. H. (1995). Asymmetric Instrument Response Resulting from Mixing Effects in Accelerated DMA-CPC Measurements, Aerosol Sci. Technol. 23:491-509. hereby incorporated by reference.) In that work, plumbing delay was accounted for by considering the transit time between DMA exit and detector, and smearing was modeled as a Continuously Stirred Tank Reactor (CSTR). A more accurate, but complicated, transfer function was obtained by combining these effects with the classical transfer function. A simpler method to account for the effect of smearing was introduced by Collins et al (See Collins, D. R., Flagan, R. C., and Seinfeld, J. H. (2002) Improved Inversion of Scanning DMA Data, Aerosol Science and technology. 36:1-9 hereby incorporated by reference herein.). In their work, transfer function determination and smearing effect were treated separately, reducing the complexity of calculations and data analysis. In both these approaches the transfer function of the scanning DMAs were assumed to be the same as that derived by Knutson and Whitby (1975) for fixed voltage DMAs.
Recently, Collins et al (See Collins, D. R., Cocker, D. R., Flagan, R. C., and Seinfeld, J. H. (2004) The Scanning DMA Transfer Function, Aerosol Science and technology. 38:833-850 hereby incorporated herein by reference.) analyzed the scanning mode DMA transfer functions obtained using simulations based on the approach of Hagwood et al. (See Hagwood, C., Sivathanu, Y., and Mulholland, G. (1999). The DMA Transfer Function with Brownian Motion a Trajectory/Monte-Carlo Approach, Aerosol Sci. Technol. 30:40-61 hereby incorporated herein by reference.) for scan times ranging from 20 to 3600 s for non-diffusive particles. For small scan times, relative to the particle residence times in the classification region, the transfer functions for scanning DMAs were shown to be substantially different from the expected triangular transfer function. Their calculation technique, using Monte-Carlo simulations, is time consuming and not amenable for real-time transfer function calculation. The recent applications of fast scan SEMS operation (e.g., Han et al., (See Han, H. S., Chen, D. R., Pui, D. Y. H., and Anderson, B. E. (2000). A Nanometer Aerosol Size Analyzer (ASA) for Rapid Measurement of High-Concentration Size Distributions, J. Nanoparticle Research 2:43-52 hereby incorporated herein by reference.) and Shah and Cocker (See Shah, S. D. and Cocker, D. R. (2005). A Fast Scanning Mobility Particle Spectrometer for Monitoring Transient Particle Size Distributions, Aero. Sci. Technol. 39:519-526 hereby incorporated herein by reference) suggests a need for near real-time transfer function calculation, accounting for the scan time of SEMS operation.
A semi-theoretical approach is introduced for near real-time transfer function calculation of scanning DMAs. Considering a cylindrical geometry and upscan operation, time-dependent particle trajectory equation is obtained and solved numerically to obtain the time-evolution of sampled concentration for a selected mobility. A simple procedure is outlined to then convert the time-spectrum of particles of one mobility to a mobility-based transfer function. The calculations suggest that fast scanning can significantly smear the transfer functions and also shift the mean mobility for a selected voltage.
A DMA consists of a coaxial cylinder with four flows, of which two flows—sheath and aerosol—are introduced at the top, and two flows—sample and excess flow—exit from bottom. An electric field of known strength is applied in the radial direction across the classification region. Charged particles coming with the aerosol flow traverse the radial direction under the influence of electric field. For a constant axial flow field, particles of different diameters (or electrical mobility) travel different distances in the classification region. High mobility particles are collected on the inner cylinder wall while low mobility particles exit with the excess flow. Particles of a narrow range of mobility are collected through the sample flow. The particles coming out of sample flow are counted using a particle counter obtaining concentrations of the classified particles. Initial size distribution measurements with the prior art DMAs were made with a stepping-mode operation, where the voltages were sequentially stepped to output particles of different mobilities. This technique, called the differential mobility particle sizing (DMPS), requires significant “dead-time” between measurements to ensure steady state aerosol sampling, and hence, accurate size distribution measurement with the DMPS requires approximately 20 minutes.
A narrow, but finite, range of mobilities exit the DMA for a selected voltage. This distribution of sampled mobilities for selected operating conditions is the instrument transfer function. For non-diffusive particles and balanced flow operation (i.e., aerosol flow equal to sample flow), the range of mobilities sampled from the DMA depends on the ratio of sample and sheath flows (Knutson and Whitby, 1975. Similar results were obtained by Stolzenburg (See Stolzenburg, M. R. (1988). An Ultrafine Aerosol Size Distribution Measuring System, Ph.D. Thesis, University of Minnesota, Minneapolis, hereby incorporated herein by reference.) following a slightly different approach. Here, we present results that are partly taken from both papers.
Following the approach of Knutson and Whitby (1975), we neglect particle inertia and diffusion and assume that the flow field is axisymmetric, laminar, and incompressible. Then the governing equations for radial and axial particle motion are:
                                                        ⅆ              r                                      ⅆ              t                                =                                    u              r                        +                          ZpE                              r                ⁢                                                                                                      ⁢                                  ⁢                                            ⅆ              z                                      ⅆ              t                                =                                    u              z                        +                          ZpE              z                                                          [        1.2        ]            
where, r is the radial distance traveled by the particle, z is the axial distance traveled by the particle, ur is the radial velocity of the particle, uz is the axial velocity of the particle, Zp is the particle mobility, Er is the electric field component in the radial direction and Ez is the electric field component in the axial direction.
The expression for stream function y and electric flux function φ can be written as
                                          Ψ            ⁡                          (                              r                ,                z                            )                                =                                    ∫                              r                ,                z                                      ⁢                          [                                                                    ru                    r                                    ⁢                                      ⅆ                    z                                                  -                                                      ru                    z                                    ⁢                                      ⅆ                    r                                                              ]                                      ⁢                                  ⁢                              φ            ⁡                          (                              r                ,                z                            )                                =                                    ∫                              r                ,                z                                      ⁢                          [                                                                    rE                    r                                    ⁢                                      ⅆ                    z                                                  -                                                      rE                    z                                    ⁢                                      ⅆ                    r                                                              ]                                                          [        1.3        ]            
Using these expressions, the substantial time derivative of ΨZpψ can be obtained as
                                                        ⅆ                              ⅆ                t                                      ⁢                          (                              ψ                +                Zpφ                            )                                =                                                    ∂                                  ∂                  t                                            ⁢                              (                                  ψ                  +                  Zpφ                                )                                      +                                                            ⅆ                  r                                                  ⅆ                  t                                            ⁢                              ∂                                  ∂                  r                                            ⁢                              (                                  ψ                  +                  Zpφ                                )                                      +                                                            ⅆ                  z                                                  ⅆ                  t                                            ⁢                              ∂                                  ∂                  z                                            ⁢                              (                                  ψ                  +                  Zpφ                                )                                                    ⁢                                  ⁢                                            ⅆ                              ⅆ                t                                      ⁢                          (                              ψ                +                Zpφ                            )                                =                      0            +                                                            ⅆ                  r                                                  ⅆ                  t                                            ⁢                              (                                                      -                                          ru                      z                                                        -                                      ZprE                    z                                                  )                                      +                                                            ⅆ                  z                                                  ⅆ                  t                                            ⁢                              (                                                      ru                    r                                    +                                      ZprE                    r                                                  )                                                    ⁢                                  ⁢                                            ⅆ                              ⅆ                t                                      ⁢                          (                              ψ                +                Zpφ                            )                                =                                    0              +                                                                    ⅆ                    r                                                        ⅆ                    t                                                  ⁢                                  (                                                            -                      r                                        ⁢                                                                  ⅆ                        z                                                                    ⅆ                        t                                                                              )                                            +                                                                    ⅆ                    z                                                        ⅆ                    t                                                  ⁢                                  (                                      r                    ⁢                                                                  ⅆ                        r                                                                    ⅆ                        t                                                                              )                                                      =            0                                              [        1.4        ]            
This derivation implies that the particles travel along trajectories in such a way that (ψ+Zpφ) remains constant. For particles of a selected mobility, the difference between two points along a trajectory implies that Δψ+ZpΔφ=0. Thus for a non-diffusive particle, the probability density function of the transfer function (ftrans) can be written as:ftrans(ψout,ψin)=δ(ψout−ψin+ZpΔφ)  [1.5]
where δ is dirac delta function. If we define ψ1 and ψ2 as stream functions at the outer and inner radii of aerosol inlet and ψ3 and ψ4 as stream function at the outer and inner radii of sample exit, then the expression for DMA transfer function (Ω) is:
                    Ω        =                              ∫                          ψ              1                                      ψ              2                                ⁢                                    (                                                ∫                                      ψ                    3                                                        ψ                    4                                                  ⁢                                                                            f                      trans                                        ⁡                                          (                                                                        ψ                          out                                                ,                                                  ψ                          in                                                                    )                                                        ⁢                                      ⅆ                                          ψ                      out                                                                                  )                        ⁢                                          f                in                            ⁡                              (                                  ψ                  in                                )                                      ⁢                          ⅆ                              ψ                in                                                                        [        1.6        ]            
where the inlet probability density function can be written as
                              f          in                =                  1                                    ψ              2                        -                          ψ              1                                                          [        1.7        ]            
For non-diffusive particles, the transfer function expression is:
                    Ω        =                              1            2                    ⁢                      1                                          ψ                2                            -                              ψ                1                                              ⁢                      (                                          -                                                                                              ψ                      4                                        -                                          ψ                      2                                        +                    ZpΔφ                                                                                +                                                                                    ψ                    4                                    -                                      ψ                    1                                    +                  ZpΔφ                                                            +                                                                                    ψ                    3                                    -                                      ψ                    2                                    +                  ZpΔφ                                                            -                                                                                    ψ                    3                                    -                                      ψ                    1                                    +                  ZpΔφ                                                                      )                                              [        1.8        ]            
The stream functions are related to the volumetric flow, which are expressed as:2π(ψ2−ψ1)=qa(aerosol)2π(ψ4−ψ2)=qsh(sheath)2π(ψ4−ψ3)=qs(sample)2π(ψ3−ψ1)=qe(excess)  [1.9]
Expressing the transfer function in the form of flow rates rather than stream function, the transfer function can be expressed as:
                    Ω        =                              1                          q              a                                ⁢                      max            ⁡                          (                              0                ,                                  min                  ⁡                                      (                                                                  q                        a                                            ,                                              q                        s                                            ,                                              (                                                                                                            1                              2                                                        ⁢                                                          (                                                                                                q                                  a                                                                +                                                                  q                                  s                                                                                            )                                                                                -                                                                                                                                                2                                ⁢                                                                  π                                  ⁢                                  Zp                                  ⁢                                  Δφ                                                                                            +                                                                                                1                                  2                                                                ⁢                                                                  (                                                                                                            q                                      e                                                                        +                                                                          q                                      sh                                                                                                        )                                                                                                                                                                                                )                                                              )                                                              )                                                          [        1.10        ]            
This expression can be further simplified in the form of dimensionless mobility and flow parameters. The dimensionless mobility is defined as:
                              Zp          _                =                              Zp                          Zp              *                                =                                                    4                ⁢                                  π                  ⁢                  LVZp                                                            (                                                      Q                    sh                                    +                                      Q                    e                                                  )                                      ⁢            log            ⁢                                          r                2                                            r                1                                                                        [        1.11        ]            
where r2 is the radii of outer cylinder, r1 is the radii of the inner cylinder, L is the length of the classification region, V is the voltage applied across classification region. The dimensionless flow parameters are:
                                                                        β                =                                                                            q                      s                                        +                                          q                      a                                                                                                  q                      sh                                        +                                          q                      e                                                                                                                          δ                =                                                    ⁢                                            q              s                        -                          q              a                                                          q              s                        +                          q              a                                                          [        1.12        ]            
Then the transfer function be written as:
                    Ω        =                              1                          2              ⁢                              β                ⁡                                  (                                      1                    -                    δ                                    )                                                              ⁢                      (                                                                                                Zp                    _                                    -                                      (                                          1                      -                      β                                        )                                                                              +                                                                                    Zp                    _                                    -                                      (                                          1                      +                      β                                        )                                                                              -                                                                                    Zp                    _                                    -                                      (                                          1                      -                      βδ                                        )                                                                              -                                                                                    Zp                    _                                    -                                      (                                          1                      +                      βδ                                        )                                                                                        )                                              [        1.13        ]            
FIG. 1 illustrates the classical DMA transfer function (Knutson and Whitby, 1975). Stolzenburg (1988) further extended the work of Knutson and Whitby by incorporating the effect of diffusion on transfer function. Hagwood et al. (1999) who used Monte Carlo approach to obtain the transfer function validated the diffusive transfer functions of Stolzenburg. In this approach, the effect of wall loss and axial diffusion in the classification region was considered. It was observed that wall loss was significant for very small particles, as their diffusion coefficient were relatively high. Also, the wall loss was seen to be much higher for plug flow than for the parabolic flow profile.
The operational regime where diffusion becomes significantly more important is estimated on the basis on resolution analysis. The DMA resolution is expressed as the ratio of mobility at the peak of the transfer function to the full width of the transfer function at the half the maximum height (Zhang and Flagan, 1996). For non-diffusive transfer function, resolution can be written as
                    R        =                                            Zp              *                                                      Δ                ⁢                Zp                            fwhm                                =                      1                          β              ⁡                              (                                  1                  +                                                          δ                                                                      )                                                                        [        1.14        ]            
For the diffusive transfer function, resolution does not have a closed form description but can be estimated from transfer functions. Flagan (1999) showed that for balanced flow, the DMA resolution is a function of the potential difference across the classification region. For typical DMA operation, it was observed that diffusion was significant for voltages <˜100 volt.
Scanning DMA:                Operating the DMA with a fixed voltage will result in the output of particles with a narrow range of mobilities, and the transfer function for this operation can be determined using the approach of Knutson and Whitby (1975), Stolzenburg (1988), and others. Stepping the operating voltages will result in size distribution measurement in 10 minutes to one hour, depending on the number of size bins and particle residence time in the DMA and the counter. Often, high temporal resolution of concentration measurements is required, for e.g., for aircraft-based measurements, smog chamber experiments, combustion processes etc. For those situations, fast operation of DMA is inevitable. For such applications, Wang and Flagan (1990) proposed to operate the DMA with a continuous variation of the electric field in the classification region. For fixed voltage operation they explained that the set of particles travelling from the aerosol inlet to the sample exit travels the same trajectory across the classification region, independent of the voltage value. Wang and Flagan (1990) showed that this behavior is preserved only for exponential variation in voltage among all possibilities of continuous variation of voltage. They also showed that this behavior is important for keeping the shape of non-diffusive transfer function independent of peak mobility (Zp*) This important observation by Wang and Flagan (1990) lead to the scanning operation of the DMA, and the resultant instrument was referred to as the scanning electrical mobility spectrometer (SEMS). This instrument was commercialized by TSI and named as scanning mobility particle spectrometer (SMPS). In scanning operation, the voltage is either increased or decreased exponentially. The expressions for voltage scan are:        
                                                        V              up                        ⁡                          (              t              )                                =                                    V              min                        ⁢                          ⅇ                              t                τ                                                    ⁢                                  ⁢                                            V              down                        ⁡                          (              t              )                                =                                    V              max                        ⁢                          ⅇ                              t                τ                                                    ⁢                                  ⁢                  τ          =                      scantime                          log              ⁡                              (                                                      V                    max                                                        V                    min                                                  )                                                                        [        1.15        ]            
Where Vup is the voltage variation with time during up scan and Vdown is the voltage variation with the time during down scan, Vmin is the minimum voltage applied across the classification region in a single scan time and Vmax is the maximum voltage applied across the classification region in a single scan time.
The scan time shown in the expression is the duration over which the voltage is changed. Using this approach the measurement time is decreased significantly to less than 5 minutes. Faster measurements with SEMS become complicated by the effect of particle concentration smearing in the counter and plumbing delays between the DMA and the counter. There have been several approaches to account for these non-idealities. The effect of smearing and plumbing delay on transfer function was considered in Russell et al. (1995). In that work, plumbing delay was accounted for by considering the transit time between DMA exit and detector, and smearing was modeled as a Continuously Stirred Tank Reactor (CSTR). A more accurate, but complicated, transfer function was obtained by combining these effects with the classical transfer function. A simpler method to account for the effect of smearing was introduced by Collins et al (2002). In their work, the transfer function determination and the smearing effect were treated separately, reducing the complexity of calculations and data analysis. In both these approaches the transfer function of the scanning DMAs were assumed to be the same as that derived by Knutson and Whitby (1975) for fixed voltage DMAs.
Recently, Collins et al (2004) analyzed the scanning mode DMA transfer functions obtained using simulations based on the approach of Hagwood et al (1999) for scan times ranging from 20 to 3600 s for non-diffusive particles. For small scan times, relative to the particle residence times in the classification region, the transfer functions for scanning DMAs were shown to be substantially different from the expected triangular transfer function. Their calculation technique using Monte-Carlo simulations, is time consuming and not amenable for real-time transfer function calculation.
In the Monte Carlo simulations done by Collins et al (2004), particles were injected from 50 different injection locations in aerosol inlet at every 5 millisecond. Particle paths were obtained by solving the trajectory equations. Particles were “collected” at 2 second time intervals at the sample exit for transfer function estimation. These simulations were done for 1000 different mobilities and transfer functions were obtained for up- and down-scan operation. The central mobility location (Zp*) obtained using their Monte Carlo simulations were seen to be different from that obtained from classical relations. The other important observation from their work was that particle trajectory between the aerosol inlet and sample exit varied as a function of tr/t. Thus, it was concluded that the variation in the transfer function is related to tr/t.
The recent applications of fast scan SEMS operation (e.g., Han et al., 2000; Shah and Cocker 2005) suggests a need for near real-time transfer function calculation, accounting for the scan time of SEMS operation. A semi-theoretical approach is introduced for near real-time transfer function calculation of scanning DMAs. Considering a cylindrical geometry and upscan operation, time-dependent particle trajectory equation is obtained and solved numerically to obtain the time-evolution of sampled concentration for a selected mobility. A simple procedure is outlined to convert the time-spectrum of particles of one mobility to a mobility-based transfer function. The calculations suggest that fast scanning can significantly smear the transfer functions and also shift the mean mobility for a selected voltage.
DMA Background and Operation:
A differential mobility analyzer (DMA) can classify particles based on electrical mobility and output particles over a narrow range of electrical mobilities. It is a standard instrument for sizing sub-micron aerosol and is commonly used in conjunction with aerosol generators to obtain monodisperse particles. DMAs typically have a cylindrical geometry, with an inner rod maintained at a selected HV and the outer cylinder at ground potential. Polydisperse aerosol is introduced near the outer cylinder and clean sheath flow is sent through the DMA between the aerosol flow and the inner rod. Charged particles will migrate across the sheath gas and be collected at distances determined by their electrical mobility. Particles of just the right size will be sampled through the sample port near the end of the central rod. The rest of the flow is pumped out as excess flow. For fast size distribution measurements over a broad particle size range, a new DMA design is being introduced here.